Introduction The equation of a quadratic function can be written in several different forms. We have practiced using the standard form of a quadratic function to identify key information and graph the function. In this lesson, we will explore writing quadratic functions in intercept form or factored form, graphing a quadratic function from intercept form, and creating the equation of a quadratic function given its graph. An equation written in intercept form allows you to easily identify the x-intercepts of a quadratic function : Creating and Graphing Equations Using the x-intercepts

Key Concepts A quadratic function in standard form can be created from the intercept form. The intercept form of a quadratic function is f(x) = a(x – p)(x – q), where p and q are the zeros of the function. To write a quadratic in intercept form, distribute the terms of the first set of parentheses over the terms of the second set of parentheses and simplify. For example, f(x) = 3(x – 4)(x – 1) becomes f(x) = 3x 2 – 15x : Creating and Graphing Equations Using the x-intercepts

Key Concepts, continued The y-intercept of a quadratic function written in intercept form can be found by substituting 0 for x. The x-intercept(s) of a quadratic function written in intercept form can be found by substituting 0 for y. The x-intercepts, also known as the zeros, roots, or solutions of the quadratic, are often identified as p and q. The axis of symmetry is located halfway between the zeros. To determine its equation, use the formula : Creating and Graphing Equations Using the x-intercepts

Key Concepts, continued Recall that the axis of symmetry extends through the vertex of a quadratic function. As such, the x-coordinate of the vertex is the value of the equation of the axis of symmetry. If the vertex is the point (h, k), then the equation of the axis of symmetry is x = h. The y-value can be found by substituting the x-value into the equation : Creating and Graphing Equations Using the x-intercepts

Common Errors/Misconceptions multiplying out the factored form when it is more advantageous not to drawing a parabola as a straight line or a series of lines instead of a curve : Creating and Graphing Equations Using the x-intercepts

Guided Practice Example 2 Identify the x-intercepts, the axis of symmetry, and the vertex of the quadratic f(x) = (x + 5)(x + 2). Use this information to graph the function : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued 1.Identify the x-intercepts and plot the points. The x-intercepts of the quadratic are the zeros or solutions of the quadratic. Start by setting the equation equal to 0 and solving to determine the zeros : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued : Creating and Graphing Equations Using the x-intercepts f(x) = (x + 5)(x + 2)Original equation 0 = (x + 5)(x + 2) Set the equation equal to 0. 0 = x + 5 or 0 = x + 2 Set each factor equal to 0. –5 = x or –2 = x Solve each equation for x.

Guided Practice: Example 2, continued 2.Determine the axis of symmetry to find the vertex. The axis of symmetry is the line that divides the parabola in half. To determine its equation, find the midpoint of the two zeros and draw the vertical line through that point. Insert the values of x into the formula to find the midpoint : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued Let p = –5 and q = –2. The equation of the axis of symmetry is x = –3.5. The axis of symmetry extends through the vertex of the quadratic. This means that the vertex (h, k) of the graph has an x-value of – : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued Substitute –3.5 for x in the original equation to determine the y-value of the vertex : Creating and Graphing Equations Using the x-intercepts f(x) = (x + 5)(x + 2)Original equation f(–3.5) = (– )(– ) Substitute –3.5 for x. f(–3.5) = (1.5)(–1.5)Simplify. f(–3.5) = –2.25

Guided Practice: Example 2, continued The y-value of the vertex is –2.25. The vertex is the point (–3.5, –2.25) : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued 3.Graph the zeros, the vertex, and the axis of symmetry of the given equation : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 2, continued 4.Sketch the function based on the zeros and the vertex : Creating and Graphing Equations Using the x-intercepts ✔

Guided Practice: Example 2, continued : Creating and Graphing Equations Using the x-intercepts

Guided Practice Example 3 Use the intercepts and a point on the graph at right to write the equation of the function in standard form : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 3, continued 1.Find the x-intercepts of the graph and write them as zeros of the equation. Identify the two points of intersection and write them as zeros to prepare to write the factors of the equation. The x-intercepts are (–2, 0) and (3, 0). The zeros are x = –2 and x = : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 3, continued 2.Write the zeros as expressions equal to 0 to determine the factors of the quadratic. x = –2x = 3 x + 2 = 0x – 3 = 0 The factors of the equation are (x + 2) and (x – 3), so the equation is f(x) = a(x + 2)(x – 3) : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 3, continued 3.Use the point (4, –3) to find a, the coefficient of the x 2 term. The equation is f(x) = a(x + 2)(x – 3) is written in intercept form, or f(x) = a(x – p)(x – q). Substitute (4, –3) into the equation to find a : Creating and Graphing Equations Using the x-intercepts

Guided Practice: Example 3, continued The coefficient of x 2 is : Creating and Graphing Equations Using the x-intercepts f(x) = a(x + 2)(x – 3)Equation –3 = a(4 + 2)(4 – 3) Substitute (4, –3) for x and f(x). –3 = a(6)(1)Simplify. –3 = 6a

Guided Practice: Example 3, continued 4.Write the equation in standard form. Insert the value found for a into the equation from step 1 and solve : Creating and Graphing Equations Using the x-intercepts f(x) = a(x + 2)(x – 3)Equation Substitute for a.

Guided Practice: Example 3, continued : Creating and Graphing Equations Using the x-intercepts Simplify by multiplying the factors. Distribute over the parentheses. The coefficient of x 2 is ✔

Guided Practice: Example 3, continued : Creating and Graphing Equations Using the x-intercepts